Historically, FPKMC algorithms have been developed to speed up KMC simulations. Indeed, the traditional KMC method may become inefficient when employed to simulate all the hops of defects on a lattice [1]} and when the transition rate matrix equation exhibits a broad spectrum of frequencies. The causes of inefficiency may be energetic or entropic in origin. In the former situation, a diffusing defect performs many transitions between a few atomic configurations connected by small energy barriers, typically a vacancy binding to a solute cluster, before escaping elsewhere [2]}. These connected configurations form trapping basins. The typical escape time of a defect is much higher than the typical time to cross into the small barriers. Subsequently, the freed vacancy may perform a considerable number of hops in bulk before recombining with another defect or being absorbed: the entropic origin for simulation inefficiency refers to this situation. The statistically exact approach to mitigate the inefficiency of the KMC methods is to draw sequences of events and first-passage times based on the theory of absorbing Markov chains [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. Mathematically, the first passage times are the sum of the residence times spent by the walker in connected states before getting absorbed by an artificial or physical sink. The absorbing sink is artificial when it corresponds to the peripheral states of an energetic trap, while a physical absorbing sink usually corresponds to an energetic trapping basin, like solute clusters and dislocations. Furthermore, once the system has reached an absorbing state in absorbing Markov chains, it stays there infinitely. Because the probability of being absorbed tends to one as time tends to infinity, the connected states of the trapping basin are commonly known as transient states. Besides, a defect initially located in any transient state can reach any absorbing state, but not necessarily in one step. The no-passage distribution is the conditional probability distribution of the defect on the transient states knowing that it has not been absorbed yet: the sum of probabilities over the transient states is one.
